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If the direction angle of vector a is 100 degrees, with a magnitude of 8, and the direction angle of vector c is 60 degrees, with a mangitude of 13, find the magnitude of resultant a + c. I know how to set this up and solve it, but for some reason I can't get the right answer.

Mathematics
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Here's what I did: |dw:1358034696496:dw| \[x = \sqrt{8^2 + 13^2 - 2(8)(13)(\cos 140°)}\]
Hmmm this is how I would do it :O So if we write our vectors in component form, we have,\[\large \vec{a}=8\cos100 \hat i+8\sin100 \hat j\]\[\large \vec c=13\cos60 \hat i+13\sin60 \hat j\]\[\large \vec a +\vec c=\color{#3366CF}{(8\cos100+13\cos60)\color{black}{\hat i}+(8\sin100+13\sin60)\color{black}{\hat j}}\] Hmm since 100 degrees isn't a nice special angle, we're going to get ugly decimal values it seems... So punch that into the calculator, then to get the MAGNITUDE of that vector we'll do uhhhh the thing, yah... I'm a little confused where your formula for x is coming from. It looks similar to the law of cosines. Maybe I just haven't done these types of problems in too long c: heh
Yeah I used law of cosines. If I add up all of the things in your method, I don't get the correct answer.

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ah ok c: lemme take another shot at it. sec.
Hmm I'm not quite sure :c In your attempt at the problem, where is the 140 degree angle coming from?
180 - 40 = 140.
(In the parallelogram)
Did you have a positive or negative value for cosine?
What did you get for the magnitude?
It would be negative since the 140 is to the left of the y axis
Sounds good. So your magnitude is greater than 13? What did you get? Is it a matter of number of decimals?
@thechocoluver445 are you still there?
It's supposed to be 9.8 but I keep getting 19. something!
I'm pretty sure my way makes sense, but I don't know.
I just realized that my way was correct. Since the resultant is opposite an obtuse angle, it must be the longest side of the triangle. Thus, 9.8 cannot be the right answer! So I was correct. Thanks to everyone for helping. :)
19.8 is correct for the problem you have posted..
The answer key was just wrong, lmao!
Yeah, that's what I got lol!
She probably just forgot the 1 before the 9.8 haha
The only thing you may want to check to see if there are typos. It is possible that they modified the question but forgot to modify the key (which is half of the current answer).
So go with 19.8.
Oh, so it's not a printed key! That make it even more probably that the answer is wrong.

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